Hermitian theta series and Maa{\ss} spaces under the action of the maximal discrete extension of the Hermitian modular group

Abstract

Let $\Gamma_n(\mathcal{\scriptstyle{O}}_{\mathbb{K}})$ denote the Hermitian modular group of degree $n$ over an imaginary quadratic number field $\mathbb{K}$ and $\Delta_{n,\mathbb{K}}^*$ its maximal discrete extension in the special unitary group $SU(n,n;\mathbb{C})$. In this paper we study the action of $\Delta_{n,\mathbb{K}}^*$ on Hermitian theta series and Maass spaces. For $n=2$ we will find theta lattices such that the corresponding theta series are modular forms with respect to $\Delta_{2,\mathbb{K}}^*$ as well as examples where this is not the case. Our second focus lies on studying two different Maass spaces. We will see that the new found group $\Delta_{2,\mathbb{K}}^*$ consolidates the different definitions of the spaces.